# In what way are (natural) numbers perfect?

In mathematics, one often makes the remark that what is being talked about is a perfect idealized object. "Our planet is a sphere, but it's not really a perfect mathematical sphere (that is perfectly smooth, etc)."

What would one say, then, about natural numbers? In what way are they perfect?

My understanding is that there might be, e.g., two somethings in front of you, but these two somethings are never really the same somethings (e.g., because they are in two different places in space at the same time, so they are not really the same object). And because arguably, the number 2 does not exist inside of space and time, 2 represents "perfect twoness", so to speak.

Is my understanding correct? If not, what would a correct answer be?

In what sense are natural numbers perfect for the mathematical platonist?

• Accoriding to SEP perfect is for their independence of humans plato.stanford.edu/entries/platonism-mathematics. I’d also say their lacking of physical properties makes them perfect, going back to Plato like nwr said. They don’t have any chaotic features of the physical world. (But we really shouldn’t say they are perfectly smooth in the physical sense either as they have no physical properties). In this sense only abstract objects are perfect, while electrons (physical) can only be perfectly independent. Jul 9 at 21:35
• Are you implying that two as observed in reality is necessarily not a "perfect (idealised) two"? I'm not sure there has to be a distinction like this. If you count two items of some kind, why can't the resulting number two be as perfect as it gets? (But then I'm not a Platonist.) Jul 9 at 22:00
• @MauroALLEGRANZA I will err on the side of Plato with this one.
– user42828
Jul 10 at 14:39
• @Ruse, yeah, I know what perfect numbers are...
– user42828
Jul 10 at 17:50
• I come from the mathematics world and am not well versed in philosophy, in particular I don't know what platonist means, but I will add what I think is the most fundamental property of the natural numbers, which is that they are (up to isomorphism) the minimally inductive set. Which loosely translates to them being the smallest infinite set Jul 13 at 7:50